Obvious statement, but how do you formally prove it?

Question

Let d be a line, M a point on the line, and n a positive integer. Why is there exactly two points at n distance from M on d? How to prove it with Euclidian axioms (without algebra) ?

Answer 1

Consider a line (say $d$ is y=0) and a point ($M$=0), then we have two directions, say positive and negative. Then we have two points at a distance $n$ from M because there are only two directions. Suppose a third point, were at a distance $n$ on the line $d$ distinct from the two previous points. Then this point must not be in the direction of the other two point (or else it would be equal to one of them) or it is not on line $d$. In both cases, such a point contradicts its definition and hence, cannot exist.

Answer 2

I think the main problem in applying Euclid's original axioms lies in combining the definition of "straight line" with that of distance (check this post).

I would use the 1:1 correspondence given in the first postulate of Birkhoff's axioms to directly find the two points corresponding to the two real numbers which are $n$ units away from the real number corresponding to $M$.